The Change of the Vibration Entropy Due to the Disordering of an Ordered Simple-Cubic Alloy

  • PDF / 262,234 Bytes
  • 7 Pages / 418.68 x 637.2 pts Page_size
  • 67 Downloads / 176 Views

DOWNLOAD

REPORT


THE CHANGE OF THE VIBRATION ENTROPY DUE TO THE DISORDERING OF AN ORDERED SIMPLE-CUBIC ALLOY

A.A.H.J. Waegmaekers and H. Bakker Natuurkundig Laboratorium der Universiteit van Amsterdam, 1018 XE Amsterdam, The Netherlands

Valckenierstraat

65,

ABSTRACT Within a model of central and noncentral vibrational entropy,

AS,

is

calculated,

forces the exact change of the

both as function of the degree of

order and as a function of the diverse force constants. A linear relationship between AS and the short-range order parameter is

found.

Results of AS as a

function of the force constants are compared with AS as calculated by an empirical formula based on a modified Einstein crystal model. INTRODUCTION On the basis of statistical mechanics several analytical approximations have been developed to describe order-disorder transformations. formation will,

fect the lattice vibrations. and this is is

Such a trans-

apart from the configurational entropy and energy,

also af-

The spectrum of lattice vibrations will change,

accompanied by a change in the vibrational entropy.

However,

it

often assumed intuitively that this contribution to the total change of

the entropy is In

negligible.

the past two attempts have been undertaken to make an estimate of the

influence of

lattice vibrations on thermodynamic properties at an order-dis-

order transformation.

Booth and Rowlinson

model of uncoupled vibrations,

[i] have shown,

using the Einstein

that the size of the discontinuity in the spe-

cific heat at the order-disorder temperature of 6-brass may be explained by taking into account the vibrational entropy. Kirkwood

[2] have attempted,

brational entropy in

Some years later Wojtowicz and

in a more elaborated way,

to incorporate the vi-

a statistical mechanical treatment of the order-disorder

transformation of a binary alloy. However,

afterwards not much work has been

done on this subject,

mainly because of the great mathematical difficulties

which are involved in

an exact calculation.

For isolated effects,

it

has been possible by using a Green's function

formalism to calculate the change in the phonon spectrum and as a consequence the change in

the vibrational entropy

(see e.g.

[3]).

However,

this method is

not suitable when a large amount of "defects" (anti-site atoms)

is

generated,

for example when an ordered alloy disorders. Recently Bakker [4,51 has developed a new numerical technique to calculate "exactly" changes in

the vibrational entropy. This method is

especially

profitable for calculations on order-disorder systems. For the one-dimensional Mat.

Res. Soc. Symp.

Proc. Vol. 21 (1984) CElsevier Science Publishing Co.,

Inc.

344

chain and the two-dimensional square lattice results have been obtained using a model of central and non-central forces [41 and a model of first

and second

nearest-neighbour forces [5]. Now results will be given for a three-dimensional simple-cubic alloy. These results will be compared with the vibrational entropy change as calculated by the simpli